# Infinite dimensional Clifford algebras?

Do infinite dimensional Clifford (and/or Grassmann) algebras exist/makes sense? Do you know good references about them?

• Infinite-dimensional exterior algebras show up as fermionic Fock spaces in second quantization, I think. – Qiaochu Yuan May 31 '14 at 17:18

There are infinite dimensional Clifford C*-algebras, which do make sense and are connect to quantum mechanics. In fact the algebra of Canonical Anticommutation Relations can be thought of as a Clifford algebra over a complex vector space, so that in field theory (or any quantum theory with infinitely many degrees of freedom) one requires Clifford algebras over infinite dimensional spaces.

Usually the algebra is defined over an infinite-dimensional space with well-defined geometric properties, namely a Hilbert space. For a real Hilbert space $H$ the Clifford algebra $Cl(H)$ is the (unique) C*-algebra generated by unitaries $u(f)$, $(f\in H)$ subject to $$u(f)u(g)+u(g)u(f)=< f,g >$$ $$u(f)^2=1, \qquad u(f+ag)=u(f)+au(g).$$ Have a look at

D. Shale, F. Stinespring "States of the Clifford algebra", Ann. Math. 80 (1964), pp 365-381

for more details and possibly other references -Ollie

Infinite dimensional clifford algebras are the setting of David Hestenes' so-called "universal geometric algebra". Hestenes uses this setting to embed vector manifolds---manifolds whose points are vectors in the UGA and thus admit a lot of niceties in terms of vector operations. For instance, if $\mathbf r(x^1, x^2, \ldots, x^n)$ is a vector function of $n$ parameters that defines a vector manifold, then the tangent vectors are indeed $\partial \mathbf r/\partial x^i$ and so on, and this is well-defined in terms of the manifold being embedded in the UGA.

For more about this application of infinite dimensional clifford algebras, see Hestens and Sobczyk's Clifford Algebra to Geometric Calculus.

Sure they make sense, and nothing really differs in the construction.

I'm not handy with the applications themselves, but I can point you to a few references I did read that hint at the applications.

• Zou. Ideal structure of Clifford algebras Advances in Applied Clifford Algebras February 2009, Volume 19, Issue 1, pp 147-153
• Wene. The idempotent structure of an infinite dimensional Clifford algebra Clifford Algebras and their Applications in Mathematical Physics Fundamental Theories of Physics Volume 47, 1992, pp 161-164

• Lounesto & Wene. Idempotent structure of Clifford algebras Acta Applicandae Mathematica July 1987, Volume 9, Issue 3, pp 165-173

My main interest was in the ring theoretic structure of the algebra itself. It's also worth noting that these are talking about the countable dimensional case only. Countable dimension covers most of the ground for applied mathematics, though :)

• Can the Clifford algebra be completed into a Hilbert space, while retaining an everywhere-defined product? (I have found that $\sup_{A,B}\frac{\lVert AB\rVert}{\lVert A\rVert\lVert B\rVert}=\infty$.) – mr_e_man Aug 8 '19 at 6:54
• @mr_e_man That would probably depend on the inner product you have in mind, but which I am not aware of. – rschwieb Aug 8 '19 at 13:46
• – mr_e_man Aug 11 '19 at 20:20
• @mr_e_man have you checked if the Clifford algebra of the completion is complete with respect to your form? That would be the first obvious thing to check. Maybe there is some categorical property that proves it easily. – rschwieb Aug 11 '19 at 22:23
• Anyway, I suppose my question should be made into a Question... – mr_e_man Aug 11 '19 at 22:35